Introduction of Monte Carlo and Quasi-Monte Carlo Simulation

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Introduction of Monte Carlo and Quasi-Monte Carlo
Simulation

• lerong_at_ee.ucla.edu

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Monte Carlo Simulation

• Problem Formulation
• Given a set of random variables X(X1, X2, Xn)T
  and a function of X, Yf(X), estimate the
  distribution of the Y
• Method
• Generate N samples of X(X1, X2, Xn)T
• For each sample of X, calculate the correspondent
  sample of Yf(X)
• Obtain the distribution of Y from the samples of
  Y

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Advantage and Disadvantage of MC simulation

• Advantage
• Accurate
• Error?0 when N?8
• Flexible
• Works for any arbitrary distribution of X
• Works for any arbitrary function of f
• Simple
• Easy to implement
• Usually used as golden case in statistical
  analysis
• Disadvantage
• Not efficient
• Need large N to obtain high accuracy
• Need to run large number of iterations
• Not suitable for statistical optimization

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Example

• Given X1 and X2 are independent standard Gaussian
  RVs, estimate the distribution of max(X1, X2)

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Quasi Monte Carlo Simulation

• Basic idea
• Use deterministic samples instead of pure random
  samples
• Select deterministic samples to cover the whole
  sample space evenly

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Discrepancy

• Definition
• N is total number of samples, A(B, P) is the
  number of points in bounding box B, ?s(B) is the
  volume of B
• Discrepancy measures how evenly the samples are
  in the sample place

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Low Discrepancy Sequence

• Sample sequence with low discrepancy
• Low discrepancy array generation algorithms
• Faure sequence
• Neiderreiter sequence
• Sobol sequence
• Halton Sequence

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Example Halton Sequence

• Basic idea
• Choose a prime number as base (let's say 2)
• Write natural number sequence 1, 2, 3, ... in
  base
• Reverse the digits, including the decimal sign
• Convert back to base 10
• 1 1.0 gt 0.1 1/2
• 2 10.0 gt 0.01 1/4
• 3 11.0 gt 0.11 3/4
• 4 100.0 gt 0.001 1/8
• 5 101.0 gt 0.101 5/8
• 6 110.0 gt 0.011 3/8
• 7 111.0 gt 0.111 7/8
• High dimensional array
• Use different base for different dimension
• Example 2-d array, X-base 2, y-base 3
• 1 gt x1/2 y1/3
• 2 gt x1/4 y2/3
• 3 gt x3/4 y1/9
• 4 gt x1/8 y4/9

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Advantage and Disadvantage of QMC Simulation

• Advantage
• Efficient
• Use fewer sample than random Monte Carlo
  simulation
• Disadvantage
• Only works in low dimension cases
• Very slow when number of random variations become
  large
• Not very common in statistical analysis

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Comparison of MC and QMC

• QMC converges faster than MC

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Simple Project

• Use Quasi MC simulation to estimate the mean of
  output delay of the following circuit
• Assume 3 variation sources with Normal
  distribution
• Leff
• Vth
• Tox
• Gate delay is linear function of variation
  sources
• Implement a Quasi-Monte Carlo sequence generator